Almost-everywhere asymptotic periodicity for dissipative outer billiards of regular polygons

For a regular polygon and a parameter $\lambda\in(0,1)$, let $T_\lambda$ denote its dissipative outer billiard map. Asymptotic periodicity conjecture. For almost every $\lambda\in(0,1)$, the map $T_\lambda$ is asymptotically periodic. Numerical experiments suggest this conjecture, while the paper notes that asymptotic periodicity is not true for every dissipative outer polygonal billiard and that a complete description of bifurcations of the $\omega$-limit set remains unavailable in general.